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Chapter 3: Problem 85
Use the One-to-One Property to solve the equation for \(x\). $$\log _{5}(x+1)=\log _{5} 6$$
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Complete the table for the time \(t\) (in years) necessary for \(P\) dollars to triple if interest is compounded annually at rate \(r\). $$ \begin{array}{|l|l|l|l|l|l|l|} \hline r & 2 \% & 4 \% & 6 \% & 8 \% & 10 \% & 12 \% \\ \hline t & & & & & & \\ \hline \end{array} $$
A logistic curve is also called a ________ curve.
A sport utility vehicle that costs $$\$ 23,300$$ new has a book value of $$\$ 12,500$$ after 2 years. (a) Find the linear model \(V=m t+b\). (b) Find the exponential model \(V=a e^{k t}\) (c) Use a graphing utility to graph the two models in the same viewing window. Which model depreciates faster in the first 2 years? (d) Find the book values of the vehicle after 1 year and after 3 years using each model. (e) Explain the advantages and disadvantages of using each model to a buyer and a seller.
Use the following information for determining sound intensity. The level of sound \(\boldsymbol{\beta}\), in decibels, with an intensity of \(I\), is given by \(\boldsymbol{\beta}=10 \log \left(I / I_{0}\right),\) where \(I_{0}\) is an intensity of \(10^{-12}\) watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 65 and 66 , find the level of sound \(\boldsymbol{\beta}\). Due to the installation of a muffler, the noise level of an engine was reduced from 88 to 72 decibels. Find the percent decrease in the intensity level of the noise as a result of the installation of the muffler.
The populations \(P\) (in thousands) of Pittsburgh, Pennsylvania from 2000 through 2007 can be modeled by \(P=\frac{2632}{1+0.083 e^{0.0500 t}}\) where \(t\) represents the year, with \(t=0\) corresponding to \(2000 .\) (Source: U.S. Census Bureau) (a) Use the model to find the populations of Pittsburgh in the years \(2000,2005,\) and 2007 . (b) Use a graphing utility to graph the function. (c) Use the graph to determine the year in which the population will reach 2.2 million. (d) Confirm your answer to part (c) algebraically.
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